[Insight-developers] BUG in BinaryBallStructuringElement
Padfield, Dirk R (GE Global Research)
padfield at research.ge.com
Thu Oct 17 09:09:49 EDT 2013
Hi Ho,
Thanks for your feedback and insight! I agree that discretizing continuous functions is always a tricky thing. Luckily, we have the spatial objects to help with this since they define their own inside-outside tests. The Ellipse spatial object is used in the BinaryBallStructuringElement implementation, but the problem is that the spatial object itself is used incorrectly. By definition, the axes should be "radius*2" rather than "radius*2+1". Defining the axes of an ellipse/circle to be "radius*2+1" is simply an error.
We can also attack this question by considering the area of the continuous function versus the discretized version by counting the number of "on" pixels in the kernel as follows:
For radius=1, the true area is pi = 3.14
Using the old version, we get 9
Using the correction, we get 5
For radius=5, the true area is 25*pi = 78.5
Using the old version, we get 97 (24% error)
Using the correction, we get 81 (3% error)
For radius=11, the true area is 121*pi = 380
Using the old version, we get 421 (11% error)
Using the correction, we get 377 (1% error)
For radius=21, the true area is 21*21*pi = 1385
Using the old version, we get 1457 (5% error)
Using the correction, we get 1373 (1% error)
As expected, as the radius increases, the discretized version better approximates the continuous function. We can also see that the corrected version is always more accurate than the old version.
What do you think?
Thanks,
Dirk
________________________________
From: Ho Cheung [hocheung20 at gmail.com]
Sent: Wednesday, October 16, 2013 6:26 PM
To: Padfield, Dirk R (GE Global Research)
Cc: ITK
Subject: Re: [Insight-developers] BUG in BinaryBallStructuringElement
Dirk,
As a counterpoint, I do not agree that there is a bug but rather just an ambiguity in the way we have defined whether or not a pixel is to be included.
If you take a protractor and plotted a unit circle, then superimpose a grid on it (this this case, 3x3), and then shaded in the nearest pixels to the circle, it would look like the “original” example. The same applies to the radius 5 circle.
Technically, if you look at the parametric definition of a circle, then yes, those pixels would not be included, as their physical space points fall outside the circle.
However, I believe (anecdotal) in graphics rendering, it is common practice to include those pixels which are nearest to the actual physical space point.
Regards,
Ho Cheung
(775) 388-2368
On Oct 16, 2013, at 1:05 PM, Padfield, Dirk R (GE Global Research) <padfield at research.ge.com<mailto:padfield at research.ge.com>> wrote:
Hi All,
I am writing to ask your advice about a bug I found in BinaryBallStructuringElement.
For a while, I have been bothered by the fact that the BinaryBallStructuringElement return balls that are larger than the specified radius. For example, when given a radius of 1, it returns the structuring element:
1 1 1
1 1 1
1 1 1
But this structuring element has a radius that is more than 1! If it truly had a radius of 1, it would be a cross shape in this case.
When choosing a larger radius, the problem is more obvious. Setting radius = 5 (leading to a structuring element size of 11x11) results in:
0 0 0 1 1 1 1 1 0 0 0
0 0 1 1 1 1 1 1 1 0 0
0 1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
0 1 1 1 1 1 1 1 1 1 0
0 0 1 1 1 1 1 1 1 0 0
0 0 0 1 1 1 1 1 0 0 0
This is clearly not an ellipse/circle with radius 5 because the interior ellipse/circle is touching each image border at five points rather than just one. As it turns out, the code is actually defining a radius that is "X + 0.5", where X is the radius that is requested!
The problem is in the specification of the ellipse axes on lines 70-76 of itkBinaryBallStructuringElement.hxx:
// Define and set the axes lengths for the ellipsoid
typename EllipsoidType::InputType axes;
for ( i = 0; i < VDimension; i++ )
{
axes[i] = this->GetSize(i);
}
spatialFunction->SetAxes(axes);
In this case, "this->GetSize()" is equal to radius*2+1. But, an ellipse/circle with radius X should have axes length 2X, not 2X+1! In the implementation, the center of the ellipse is properly accounted for by setting it to "this->GetRadius+1", but the size of the ellipse is not correct!
To correct this, we can make a simple change either
axes[i] = this->GetSize(i) - 1;
or
axes[i] = this->GetRadius(i) * 2;
The second is probably more intuitive.
With this fix, we get for radius=1:
0 1 0
1 1 1
0 1 0
and for radius=5:
0 0 0 0 0 1 0 0 0 0 0
0 0 1 1 1 1 1 1 1 0 0
0 1 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 1 0
1 1 1 1 1 1 1 1 1 1 1
0 1 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 1 0
0 1 1 1 1 1 1 1 1 1 0
0 0 1 1 1 1 1 1 1 0 0
0 0 0 0 0 1 0 0 0 0 0
This is a true circle with radius 5!
My questions are:
1) Is anyone else bothered by this bug? I imagine that many users expect the corrected version and don't realize they are getting the incorrect one.
2) Do others agree with this fix?
3) Can we make this fix given the number of filters/applications that will change slightly as a result of this fix?
Many thanks,
Dirk
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